The duchess was admitted on Monday after developing a condition known as hyperemesis gravidarum, which leaves expectant mothers so ill they cannot keep food or liquids down.

But she looked well and even smiled briefly as she posed with Prince William on the steps of the King Edward VII hospital for the world’s media.

Prince Williams and the Duchess of Cambridge leave the King Edward VII hospital in central London.

Photo: AFP

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When a member of the press asked how she was feeling, the Duchess, wrapped up against the cold in a coat and scarf and carrying a bouquet of yellow flowers, replied: ‘‘Much better.’’

The Prince of Wales said on Thursday he was ‘‘thrilled’’ that the duchess was pregnant, adding: ‘‘It’s a very nice thought to become a grandfather in my old age.’’

Kate, who is less than 12 weeks pregnant, will now have a period of rest at her home at Kensington Palace, said St James’s Palace.The duchess’s attendance at two royal engagements this weekend - a Centrepoint Royal Albert Hall fundraiser and the British Military Tournament - has been cancelled.

And it is likely her public diary will be tailored to how she is coping with the illness that causes prolonged bouts of vomiting triggered by smells and even movement.

Her discharge comes the day after it emerged two Australian radio DJs impersonated the Queen and the Prince of Wales to dupe hospital staff into giving a condition update on the Duchess on Tuesday.

The 2Day FM presenters, Mel Greig and Michael Christian, apologised for their actions, as did their radio station.

But Christian has been promoting their stunt on Twitter, telling followers in a tweet: ‘‘Still haven’t heard the #RoyalPrank that has the world talking? Listen to it here...’’

According to reports, the Australian Communications and Media Authority has received complaints about the prank call, but said they should first be made to the licensee.

The presenters remarked during their show how their efforts were the ‘‘easiest prank call ever made’’, as they put on mock British accents they later described as ‘‘terrible’’.